The boos rang around Valley Parade at the final whistle as Bradford's miserable form in front of their own supporters continued.
In a game of relatively few chances Tinkler's 34th minute strike from Richie Humphreys' corner was to prove as decisive as it was controversial.
The Bradford players were not happy that the referee deemed Dean McDonald's wild effort had taken a slight deflection and the protests became louder after Hartlepool took the lead from the resulting coroner with Tinkler rising at the back post to nod home.
City thought they had equalised virtually straight away but a superb Dean Windass volley was ruled out for offside.
Steve Schumacher's delightful chip found City's leading scorer in space and the veteran frontman thought that he had ended his goal drought but the linesman's flag cut short the celebrations.
Although the front pairing of Windass and Danny Cadamarteri looked lively, Bradford's impotent midfield lacked the guile to break down a stubborn Hartlepool side.
Home boss Colin Todd brought on winger Owen Morrison at half time to add some much needed craft to a lacklustre midfield.
Although Bradford enjoyed the lion's share of possession in the second period they failed to test Dimitrios Konstantopoulos in the Hartlepool goal.
The closest they came was a Lewis Emanuel volley that was bravely blocked by Darren Williams.
The win lifts Martin Scott's improving side above Bradford in the table.